In this paper, a new algorithm for Sparse Component Analysis (SCA) or atomic decomposition on over-complete dictionaries is presented. The algorithm is essentially a method for obtaining sufficiently sparse solutions of underdetermined systems of linear equations. The solution obtained by the proposed algorithm is compared with the minimum 1-norm solution achieved by Linear Programming (LP). It is experimentally shown that the proposed algorithm is about two orders of magnitude faster than the state-of-the-art 1-magic, while providing the same (or better) accuracy.